A380406 -id:A380406 - cp's OEIS frontend

A380425 E.g.f. A(x) satisfies A(x) = exp( 2 * x * A(x)^(1/2) * exp(x * A(x)^(1/2)) ).

1, 2, 12, 116, 1592, 28472, 630028, 16649348, 512197456, 17993496176, 711065689364, 31231930472492, 1509776777566648, 79670350504209896, 4557716010219325468, 280992142281969312548, 18574365176584473753248, 1310583528463442480750048, 98318677221689347734929956

Offset: 0

Seiichi Manyama, Jan 24 2025

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..360

Cf. A162695, A380426, A380427.

Cf. A360474, A380406.

a(n) = 2*sum(k=0, n, k^(n-k)*(n+2)^(k-1)*binomial(n, k));

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A162695.

a(n) = 2 * Sum_{k=0..n} k^(n-k) * (n+2)^(k-1) * binomial(n,k).

A380407 E.g.f. satisfies A(x) = exp( 3 * x * exp(x) * A(x)^(1/3) ).

Original entry on oeis.org

1, 3, 21, 207, 2697, 43803, 854685, 19512615, 510977937, 15112457523, 498560461989, 18160560320895, 724240913035545, 31394996915447883, 1470245245400432685, 73987438021589516247, 3982389565847576723745, 228331703268783136636515, 13894569264190369648271157

Offset: 0

Seiichi Manyama, Jan 23 2025

nonn

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A273954, A357247, A380406.

terms = 19; A[] = 0; Do[A[x] = Exp[3*x*Exp[x]*A[x]^(1/3)] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x]Range[0,terms-1]! (* Stefano Spezia, Jun 14 2025 *)

a(n) = 3*sum(k=0, n, k^(n-k)*(k+3)^(k-1)*binomial(n, k));

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A273954.
E.g.f.: A(x) = exp( -3*LambertW(-x * exp(x)) ).
a(n) = 3 * Sum_{k=0..n} k^(n-k) * (k+3)^(k-1) * binomial(n,k).

cp's OEIS Frontend

A380425 E.g.f. A(x) satisfies A(x) = exp( 2 * x * A(x)^(1/2) * exp(x * A(x)^(1/2)) ).

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